Abstract
In this chapter we introduce the analysis of boundary integral operators on a polygon with the tool of the Mellin transformation from the original paper [128]. The interested reader may also look into [241] where the Mellin calculus is used to analyse the mapping properties of the integral operators in countably normed spaces. These results are crucial for deriving exponentially fast convergence of the hp −version of the boundary element method (see Chap. 8). The results of the subsection describing the regularity of the solution near the vertices were originally published in [138]. The Mellin calculus is used in Sect. 9.3 to analyze the regularity of the solution at the tip of an interface crack. In Sect. 9.4 to analyze the mixed boundary value problem for the Laplacian with the hyper singular operator and the singular behaviour of its solution at the point where Dirichlet and Neumann conditions meet and in Sect. 9.5 to analyze the mapping propeties of boundary integral operators with countably normed spaces. In the frame work of the spaces the analysis of the exponential convergence of the hp Galerkin approximation is presented in Sect. 8.1.
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